**Question:**

$X_1 , X_2 , ... X_n$ are unif(0, 1) random variables and 
$Y_n = {\sqrt n} \min \{{\sqrt X_1}, {\sqrt X_2}, ... {\sqrt X_n}\} $
Consider the sequence $Y_1 , Y_2 , ... Y_n$ 
and give the pdf or pmf of the limiting distribution, if it exists

**My attempt:**

The Support of $Y_n$ is $(0, {\sqrt n})$  , so for $ y \le 0$ $F_{Y_n}$ $(y) = 0$ and for $y \ge {\sqrt n}$ we have $ F_{Y_n}$ $(y) = 1$.

Then for $y \in (0, {\sqrt n})$ we have: 

$F_{Y_n}$ $(y) = P(Y_n \le y) \\ =P({\sqrt n} \min \{{\sqrt Y_1}, {\sqrt Y_2}, ... {\sqrt Y_n}\} \le y) \\ =P(\min \{{\sqrt Y_1}, {\sqrt Y_2}, ... {\sqrt Y_n}\} \le y/{\sqrt n}) \\ =[1 - (1 - (y/{\sqrt n}))]^n$

which gives you the cdf:
  $F_{Y_n}$ $(y) =
\begin{cases}
1,  & y \ge {\sqrt n} \\
[1 - (1 - (y/{\sqrt n}))]^n, & 0 \lt y\lt {\sqrt n} \\
0, & \ y\le 0
\end{cases}$  

Then: $\lim_{n\to\infty} F_{Y_n} (y) =
\begin{cases}
1 - e^{-y},  & y \gt 0 \\
0, & \ y\le 0
\end{cases}$  

And the pdf is then:
$f_{Y_n} (y) = e^{-y}$ where $y \ge 0$

Is this correct?  Not very confident in what I have done.